Global patterns in signed permutations
Owen John Levens, Joel Brewster Lewis, Bridget Eileen Tenner
TL;DR
This work studies global pattern avoidance in signed permutations $S^B_n$, showing that many structural properties of these objects can be characterized by the same global avoidance sets as in the symmetric group, a phenomenon termed persistence. Leveraging the rc-invariant inclusion map $ ext{iota}$ and the finite conversion set $ ext{gl}(P)$, the authors connect global patterns to classical patterns and employ the Garfinkle--Barbasch--Vogan (GBV) correspondence with domino tableaux to translate global avoidance into tableau constraints, enabling both characterization and enumerative results. They prove persistence for vexillarity, boolean, freeness, and, in a weak sense, smoothness and full commutativity, while identifying properties (e.g., Grassmannian, bigrassmannian, rank-symmetry) that do not persist. On the enumerative side, they extend Egge’s results, examining $| ext{GAV}_n(P)|$ for short patterns, and present two novel proofs for $| ext{GAV}_n(321)|=inom{2n}{n}$, tying global counts to central binomial coefficients via domino tableaux and bijections. The findings illuminate the deep connections between global pattern avoidance, rc-invariance, and classical type-A phenomena, with implications for Coxeter combinatorics and related algebraic geometry via Schubert varieties and the Erdős–Szekeres paradigm.
Abstract
Global permutation patterns have recently been shown to characterize important properties of a Coxeter group. Here we study global patterns in the context of signed permutations, with both characterizing and enumerative results. Surprisingly, many properties of signed permutations may be characterized by avoidance of the same set of patterns as the corresponding properties in the symmetric group. We also extend previous enumerative work of Egge, and our work has connections to the Garfinkle--Barbasch--Vogan correspondence, the Erdős--Szekeres theorem, and well-known integer sequences.